Eckmann-Hilton duality

Statement

Suppose $S$ is a set and $*$ and $\cdot$ are binary operations on $S$ satisfying:

$\!(a*b)\cdot (c*d)=(a\cdot c)*(b\cdot d)\ \forall a,b,c,d\ \in S$.

Note that this condition is symmetric in $*$ and $\cdot$, and can be interpreted as saying that $\cdot$ is a homomorphism from the magma $(S\times S,*\times *)$ to $(S,*)$. In the special case where $*=\cdot$, we get what is called a medial magma.

Suppose we have the following additional hypothesis:

Hypothesis: There exists $e\in S$ that is a two-sided neutral element for $*$ as well as for $\cdot$.

Then, we obtain the conclusions:

• $*=\cdot$.
• The common operation is commutative.